Geodesic
An example of the decomposition non-uniqueness for a 3-dimensional geometric object
Čelâbinskij fiziko-matematičeskij žurnal, Tome 4 (2019) no. 3, pp. 265-275 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In 1942, M.H.A. Newman formulated and proved a simple lemma that has been very useful in various areas of mathematics in particular in algebra and Gröbner — Shirshov bases theory. It was later called Diamond Lemma, since its key design is graphically depicted as a rhombus (diamond symbol). In 2005, I proposed a new version of this lemma, designed to solve geometric problems, and proved existence and uniqueness theorems for primary decompositions of various geometric objects: 3-dimensional manifolds, knots in thickened surfaces, knotted graphs, knotted theta curves in 3-dimensional manifolds. It turned out that all geometric objects of the mentioned types allow primary decomposition, but in some cases (for example, for orbifolds) uniqueness decomposition is absent. This article presents this new version of the lemma and an algorithm for its application. I propose a theorem that uses Diamond Lemma to prove it, and a counterexample showing the impossibility of omitting one of the conditions of the theorem.
Keywords: 3-dimensional manifold, knot, knotted graph, Diamond Lemma, prime decompositions of geometric objects. 
@article{CHFMJ_2019_4_3_a1,
     author = {S. V. Matveev},
     title = {An example of the decomposition non-uniqueness for a 3-dimensional geometric object},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {265--275},
     year = {2019},
     volume = {4},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2019_4_3_a1/}
}
TY  - JOUR
AU  - S. V. Matveev
TI  - An example of the decomposition non-uniqueness for a 3-dimensional geometric object
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2019
SP  - 265
EP  - 275
VL  - 4
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2019_4_3_a1/
LA  - ru
ID  - CHFMJ_2019_4_3_a1
ER  - 
%0 Journal Article
%A S. V. Matveev
%T An example of the decomposition non-uniqueness for a 3-dimensional geometric object
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2019
%P 265-275
%V 4
%N 3
%U http://geodesic.mathdoc.fr/item/CHFMJ_2019_4_3_a1/
%G ru
%F CHFMJ_2019_4_3_a1
S. V. Matveev. An example of the decomposition non-uniqueness for a 3-dimensional geometric object. Čelâbinskij fiziko-matematičeskij žurnal, Tome 4 (2019) no. 3, pp. 265-275. http://geodesic.mathdoc.fr/item/CHFMJ_2019_4_3_a1/

[1] M. H. A. Newman, “On theories with a combinatorial definition of "equivalence" ”, Annals of Mathematics, 43 (1943), 223–243 | DOI | MR

[2] Shirshov A.I., “Some algorithmic issues for Lie algebras”, Siberian Mathematical Journal, 3:2 (1962), 292–296 (In Russ.) | Zbl

[3] Bokut' L.A., “Embeddings into simple associative algebras”, Algebra and Logic, 15:2 (1976), 73–90 (In Russ.) | MR | Zbl

[4] G. M. Bergman, “The diamond lemma for ring theory”, Advances in Mathematics, 29 (1978), 178–218 | DOI | MR

[5] T. Becker, V. Weispfenning, Gröbner Bases: A Computational Approach to Commutative Algebra, Springer-Verlag, New York, 1998, 576 pp. | MR

[6] H. Kneser, “Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten”, Jahresbericht der Deutschen Mathematiker-Vereinigung, 28 (1929), 248–260

[7] H. Schubert, “Die eindeutige Zerlegbarkeit eines Knotens in Primknoten”, S. B. Heidelberger Akad. Wiss. Math.-Natur. Kl., 3 (1949), 57–104 | MR

[8] K. Miyazaki, “Conjugation and prime decomposition of knots in closed, oriented $3$-manifolds”, Transactions of the American Mathematical Society, 313 (1989), 785–804 | MR | Zbl

[9] S. Matveev, V. Turaev, “A semigroup of theta-curves in 3-manifolds”, Moscow Mathematical Journal, 4 (2011), 1–10 | MR